Proceedings of the 3rd International Conference on Fluid Flow, Heat and Mass Transfer (FFHMT’16)
Ottawa, Canada – May 2 – 3, 2016
Paper No. 130
Large Eddy Simulation of Confined Turbulent
Round Jet with Annular Jets
Christian Lagarza, Martín Salinas, Eliseo Martínez, Jorge Ramírez
Ciudad Universitaria, Engineering Institute/UNAM
Av. Universidad 3000, Circuito Exterior S/N Delegación Coyoacán, C.P. 04510, México City, México
[email protected]; [email protected]; EMartí[email protected]
Abstract – Particular geometry is used to simulate numerically a confined turbulent jet. The purpose of this work is evaluate
numerically the effect of annular jets above a central jet and define features of turbulence by analysis of anisotropy turbulence. Threedimensional large eddy simulation was used for turbulence modelling. Lumley-Newman triangle representation of anisotropy
turbulence was used to define differences between numerical measures of local turbulence and 3D isotropic turbulence state.
Keywords: LES, round jet, annular jet, confined, turbulence, anisotropy.
1. Introduction
In many industrial applications, mixtures are produced by jets. Generally this kind of flow has two features, the
turbulent flow and it is confined. Considering the effect of the surrounding enclosure or confinement a jet presents
significant behavior changes. Unlike the free jet sufficient amount of surrounding fluid is not present for the entrainment in
confined jet [1]. This develop a recirculation flow pattern. A free jet and confined jet, are quite different (Figure1). In
addition the pipe diameter ratio (𝐷𝑑 ⁄𝐷𝑗 ) present another important feature. Diameter ratios lower than 11 refers to a strong
confinement [2].
Fig. 1: Difference between an array of free turbulent jets and confined turbulent jets.
Although there are studies about the turbulence of confined jets [3, 4, 5, 6, 7], information related with strong
confined turbulent jet when it is affected by annular jets is null. Therefore, the purpose of this work is to analyse the effect
of confinement and peripheral jets about the turbulence of main jet. This becomes relevant when process tend to be
performed in confined or restricted spatial regions.
2. Governing equations
In a frame of Cartesian reference compressible Navier-Stokes equations can be written in the form
𝜕𝑈 𝜕𝐹𝑖
+
=𝑆
𝜕𝑡 𝜕𝑥𝑖
(1)
𝑈 is a vector of five component defined by 𝑈 = (𝜌, 𝜌𝑢1 , 𝜌𝑢2 , 𝜌𝑢3 , 𝜌𝑒)𝑇 . It also considers that 𝑢 = (𝑢1 , 𝑢2 , 𝑢3 ) is the
velocity vector and 𝜌 is the density. Equation 1 shows the evolution of the density, momentum and total energy. 𝑒 is
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1
defined as an ideal gas by two terms. First, 𝐶𝑣 𝑇 of internal energy and second, (𝑢12 + 𝑢22 + 𝑢32 ) of kinetic energy.
2
𝐹𝑖 are fluxes where ∀ 𝑖 ∈ {1, 2, 3} and for a Newtonian fluid are given by
𝐹𝑖 =
𝜌𝑢𝑖
𝜌𝑢𝑖 𝑢1 + 𝜌𝛿𝑖1 − 2𝜇𝑆𝑖1
𝜌𝑢𝑖 𝑢2 + 𝜌𝛿𝑖2 − 2𝜇𝑆𝑖2
𝜌𝑢𝑖 𝑢3 + 𝜌𝛿𝑖3 − 2𝜇𝑆𝑖3
(2)
𝜕𝑇
(𝜌𝑒 + 𝑝)𝑢𝑖 − 2𝜇𝑢𝑗 𝑆𝑖𝑗 − 𝑘
[
𝜕𝑥𝑖 ]
𝑘 = 𝜌𝐶𝑝 𝐾 is the thermal conductivity and 𝐾 is the thermal diffusivity. The symbol 𝛿𝑖𝑗 is the Kronecker delta and
𝑆𝑖𝑗 is the deviatoric component of strain tensor. The term 𝑆𝑖𝑗 is written as
𝑆𝑖𝑗 =
1 𝜕𝑢𝑖 𝜕𝑢𝑗 2
(
+
− (∇ ∙ 𝑢)𝛿𝑖𝑗 )
2 𝜕𝑥𝑗 𝜕𝑥𝑖 3
(3)
Molecular viscosity is established through empirical Sutherland law. The thermal conductivity K (T) is
obtained assuming that the molecular Prandtl number is 0.7.
3. Large eddy simulation
Equations are closed from Large Eddy Simulation technique. It means simulate directly all large scale in the flow
without need a turbulence model. At the same time, these large scales will be filtered larger in a local mesh size and its
effect on the movement of small scales are modelled after with a sub-mesh model. For filtered equations and sub-mesh
terms, see [8]. The sub-grid scale model used is the selective structure function model, which is an extrapolation to the
physical space of the spectral model [9].
3.1. Numerics
The code used in this work was developed in Engineering Institute of UNAM and it has already been tested to
simulate different turbulent flows [8, 9, 10, 11]. This in-house code use a generalized coordinate system and is solved by
an extension of the explicit scheme McCormack, second order in time and fourth order in space, developed by Gottlieb and
Turkel. Adaptation to the generalized coordinates is performed by introducing a Jacobian matrix. It transforms a complex
or curvilinear geometry in a Cartesian coordinate system mesh in a simple orthogonal geometry with uniform mesh. This
scheme is a predictor-corrector scheme.
3.2. Flow configuration, mesh and boundary conditions
The geometry is similar to inverse diffusion flame burner. It only was considered like a mixing chamber and
simulated as non-reactive flow. Central round jet is injected in a quiescent environment, Figure 2. At the same time four
annular jets inject fluid into the chamber. Central jet remain at constant velocity (𝑈𝑗 ) and annular round jets were tested
with different velocities (𝑈𝑎 ). Velocity ratios (𝑈𝑎 ⁄𝑈𝑗 ) were 0.5, 1.0 and 2.0 times of main jet velocity (Table 1).
Fig. 2: Geometry and computational domain of mixing chamber.
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Table 1: Simulated cases.
Velocity ratios (𝑼𝒂 ⁄𝑼𝒋 )
0.5
1.0
2.0
Case
C1
C2
C3
Reynolds number = 10800, the Mach number = 0.3 (subsonic flow) and the Prandtl number = 0.7. Only a 12
diameters on axial direction of the chamber is simulated to save time calculation. Injection has a top-hat profile. Boundary
conditions proposed by [12] are applied and used with fixed temperature and velocity at inlet, and prescribed pressure for
outlet. Walls are considered as adiabatic walls.
4. Code validation
The validation code was performed using a baseline case (BC). BC represent single round turbulent jet. Numerical
results were compared with the experimental data of [3]. After initial and transitional region, the flow develops selfpreserving behavior. Self-similarity region was analysed. The inverse of velocity profile along the axis present a linear
behaviour.
𝑈0
1 𝑥 − 𝑥0
(
)
=
𝑈𝑐 𝐵𝑒
𝐷
(4)
𝑥
𝑈0 is exit velocity, 𝑈𝑐 is centreline mean velocity and 0 ≈ 4, where 𝑥0 is the virtual origin. Slope value of decrease
𝐷
of velocity in self-similar region is presented in the Table 2. Although there is a change in slope it shows a good agreement
with experimental data. Also, velocity and steamwise turbulence intensity profiles were compared, Figure 3.
Table 2: Centreline mean velocity parameter, 1⁄𝐵𝑒.
Hussein (1994)
0.1724
Hussein (1994)
Present work
0.1743
U/Uc
U0/Uc
Self-similar region
1/Be
Uo/Uc LES
z/nD
Hussein (1994)
z/5D
z/6D
z/7D
z/8D
w'/Uc
u'/Uc
x/D
Hussein (1994)
z/5D LES
z/6D LES
z/7D LES
z/8D LES
z/9D LES
z/nD
Hussein (1994)
z/5D
z/6D
z/7D
z/8D
z/nD
Fig. 3: Centreline inverse mean velocity profile (top-left), steamwise mean velocity profile (top-right), steamwise turbulence intensity
profile (Bottom-left) and radial turbulence intensity (bottom-right).
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5. Results
5.1. Vortex structures
As a first analysis vortex structures visualization was performed using 𝑄 criterion, Figure 5. For BC, when the
jet leave the nozzle amplified perturbation are developed. Furthermore, the entrain fluid received and spread in radial
direction downstream produce an increase on size until the initial momentum is dissipated by viscous effects [13]. In
Figure 5, in C1 the peripheral jets cause a change in the vortex zone size within the first-half (axial length) of the
central jet, this is due to turbulent interactions. Additionally, in the second-half, no modifications are observed.
Turbulent region conserve similar behaviour as BC. In the following cases (C2, C3) an increase in the velocity also
produce an increase in turbulence intensity of peripheral jets. C3 present a particular feature, the increase in the
intensity causes an apparent contraction that affects the development of the central flow.
BC
C1
C2
C3
Fig. 5: Iso-surface of 𝑄 criterion, 𝑄 = 30.0.
An instantaneous iso-contour of concentration is presented in Figure 6. All cases present coherent structures at
outside of the jet. However, cases with peripheral jets shows earlier transition to turbulence and mixing.
Consequently, C3 has a shorter potential core, Figure 7. The difference between the main jet and the annular jet
velocity seems to aid the development of turbulence and affects the mixing efficiency.
BC
C1
C1
C2
C3
C3
Fig. 6: Snapshot of concentration in the middle section of axial length (left) and iso-surface of concentration 𝑐 = 0.5 with
contour of the magnitude of velocity for C1 and C3 (right).
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5.2. Anisotropy Invariant Map (AIM)
To characterize the main jet turbulence state, Lumley-Newman representation was used [14, 15, 16], it is based on
the use of anisotropic invariant of the Reynolds stress tensor (𝑎𝑖𝑗 ), and is obtained by deviatoric tensor (𝑏𝑖𝑗 )
̅̅̅̅̅̅̅
𝑏𝑖𝑗
𝑢𝑖 ′𝑢𝑗 ′ 2
(5)
𝑎𝑖𝑗 =
=
− 𝛿𝑖𝑗
𝑘𝑒𝑐
𝑘𝑒𝑐
3
where 𝑘𝑒𝑐 represents turbulent kinetic energy and 𝑢′ are the rms-fluctuations of the velocity. This tensor is the basis of
turbulence anisotropy analysis. With the second (𝐼2) and third (𝐼3) invariant defined as follow
𝐼2 = 𝑎𝑖𝑗 𝑎𝑗𝑖
(6)
𝐼3 = 𝑎𝑖𝑗 𝑎𝑗𝑘 𝑎𝑘𝑖
(7)
the Lumley triangle is established to determine the degree of anisotropy and its nature. For analysis, anisotropy invariant
maps were obtained from three different regions along the axial length for each case, Figure 7.
Z1
Z2
Z3
Fig. 7: Analyzed zones: Z1 - initial, Z2 – middle and Z3 – final.
Z1
Z2
Fig. 8: Anisotropy invariant maps of analysed zones.
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Z3
Results of all analysed zones are presented in Fig. 8. In Z3, near of the outlet chamber: C2 is not significantly
affected by the peripheral jets and behaves similar to BC. In BC turbulence has not a well-defined state. Points are
scattered throughout the triangle area. However, the state tends to be axisymmetric of two components. In C3, the
behaviour changes from two components to a single component. This implies that the peripheral jets influence the state of
turbulence downstream of the chamber (the potential core is also affected).
In Z1y Z2, C2 is not influenced significantly by the peripheral jets. However, turbulence is more homogeneous and
isotropic. This case tends to an axisymmetric behaviour of two components with a form of "cigarette". This behaviour is
common in free shear layers and jets. In contrast, in C3, turbulence sate passes of two components to a single component.
This means that peripheral jets generate significant decrease in the radial fluctuations. The annular jets act as walls over
central jet.
4. Conclusion
Three strongly confined turbulent round jet with four annular jets have been studied by LES and anisotropy invariant
map. The velocity ratios range from 0.5 to 2.0. In this work, the effect of annular jets over central jet, the degree of
anisotropy and the nature of turbulence is analysed.
Velocity ratios greater or equal to 1 intensify turbulent region in the stream. This causes increased interaction
between the central jet and annular jets. Analysis shows reduction of potential core in central jet when the peripheral jets
have higher velocities than the central jet velocity. However, peripheral jets generate a significant decrease in radial
fluctuations. It means that the annular jets act as walls and confine the central jet. This could adversely affect the mixing
process.
Acknowledgements
The authors appreciate the financial support provided by National Autonomous University of Mexico, PAPIIT
DGAPA IA103515. The authors gratefully acknowledge the support from Ricardo Del Rio Serrano, Jonathan Hernández
García, Ana Karen Moreno Álvarez and Martha Luisa Preciado Méndez.
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